We have,
\(\widetilde{\Gamma }_{\mathfrak {p}}^{d} = \bigsqcup _{b+\mathfrak {p}\in \mathfrak {o}/ \mathfrak {p}} \sigma _b^{-1}\left((\mathfrak {p} \setminus \mathfrak {p}^2 )\times \mathfrak {p}^{d-1} \times \mathfrak {o}\setminus \mathfrak {p}\right) = \bigsqcup _{b+\mathfrak {p}\in \mathfrak {o}/ \mathfrak {p}} \sigma _b^{-1}\left({\Gamma _\mathfrak {p}}^d\right) , \) 
where \(b+\mathfrak {p}\)  is an element in \(\mathfrak {o}/\mathfrak {p}\) .
Let \(U_\mathfrak {p}\)  be defined as before.
We set
\(\widetilde{U}_\mathfrak {p}= \bigsqcup \limits _{b+\mathfrak {p}\in \mathfrak {o}/ \mathfrak {p}} \sigma _b^{-1} \left(U_\mathfrak {p}\right) \subset {\mathfrak {o}_\mathfrak {p}}^{d+1},\) 
and \(\widetilde{U}_\infty = \emptyset \) .
As noted before, \(\sigma _b(\mathfrak {o}^{d+1})= \mathfrak {o}^{d+1}\)  for any \(b\in \mathfrak {o}\)  and thus, we have \(\widetilde{\Gamma }_\mathfrak {p}^d = \mathfrak {o}^{d+1} \cap \widetilde{U}_\mathfrak {p}\) .
Now, with the system of sets \(\widetilde{U}_\mathfrak {p}\)  and the map \(P\)  defined as in Theorem REF , we have, as before, that \(P^{-1}(\lbrace \emptyset \rbrace )^C\)  is the set \(\widetilde{\Gamma }^d\)  of shifted Eisenstein polynomials of degree \(d\) . Since \(\mathfrak {o}_\mathfrak {p}\)  is compact, and \({\sigma _b}\)  is a surjective and continuous endomorphism of \(\mathfrak {o}_\mathfrak {p}^{d+1}\) , we have that for any \(b \in \mathfrak {o}\) , \(\sigma _b\)  preserves the Haar measure. Thus,
\(\nu _{\mathfrak {p}}(\widetilde{U}_\mathfrak {p}) = \nu _\mathfrak {p}\left(\bigsqcup _{b+\mathfrak {p}\in \mathfrak {o}/ \mathfrak {p}} \sigma _b^{-1}\left(U_\mathfrak {p}\right)\right) = \left|{\mathfrak {o}}/{\mathfrak {p}}\right|\nu _{\mathfrak {p}}({U}_\mathfrak {p})= p^{\deg (\mathfrak {p})}\cdot \nu _{\mathfrak {p}}({U}_\mathfrak {p}).\) 
We first deal with the case \(d\ge 3\)  and verify condition (REF ) for the system \((\widetilde{U}_\eta )_{\eta \in N_K}\) . For this, we define polynomials \(F, G \in \mathfrak {o}[x_0, \ldots , x_{d}]\)  as
\(F(x_0, \ldots , x_{d})& = d^2 x_0 x_{d} - x_1 x_{d-1},\\G(x_0, \ldots , x_{d})& = x_2 x_{d-2} - \binom{d}{2}^2 x_0 x_{d}.\) 
 Clearly, these are coprime for any value of \(d\ge 3\) . We claim that
\( \mathfrak {o}^{d+1} \bigcap \left(\bigcup \limits _{\mathfrak {p}\succ M} \widetilde{U}_\mathfrak {p}\right) = \bigcup \limits _{\mathfrak {p}\succ M} \widetilde{\Gamma }_\mathfrak {p}^d \subseteq S_M(F,G).\) 
Let \( a^\prime =(a_0^\prime , a_1^\prime , \ldots , a_{d}^\prime ) \in \widetilde{\Gamma }_\mathfrak {p}^d\)  for some \(\mathfrak {p}\succ M\) . Then, there exists \(b \in \mathfrak {o}\)  such that \(\sigma _{-b}(a^\prime )=a=(a_0, \dots , a_d)\)  is \(\mathfrak {p}\) -Eisenstein.
This is equivalent to saying \(a^{\prime }=\sigma _b(a)\)  and hence we have the relations \(a_j^\prime = \sum _{i=j}^d \binom{i}{j}a_i b^{i-j}\) , \(a_i \equiv 0 \mathfrak {}\mod {\mathfrak {p}}\)  for \(i\in \lbrace 0, \dots , d-1\rbrace \)  and \(a_d \notin \mathfrak {p}\) . Thus, \(a_j^\prime = \binom{d}{j}a_d b^{d-j} \mathfrak {}\mod {\mathfrak {p}}\) . Clearly, \(a_j^\prime a_{d-j}^\prime \equiv \binom{d}{j}^2 a_d^2 b^d \mathfrak {}\mod {\mathfrak {p}}\) , and in particular
\(a_0^\prime a_{d}^\prime &\equiv a_d^2 b^d \mathfrak {}\mod {\mathfrak {p}}, \\a_1^\prime a_{d-1}^\prime & \equiv a_d^2 b^d d^2 \mathfrak {}\mod {\mathfrak {p}}, \\ a_2^\prime a_{d-2}^\prime & \equiv a_d^2 b^d \binom{d}{2}^2 \mathfrak {}\mod {\mathfrak {p}}.\) 
 Therefore, we get
\(F(a^\prime ) \equiv G(a^\prime ) \equiv 0 \mathfrak {}\mod {\mathfrak {p}}\) 
or in other words \(a^\prime \in S_M(F,G)\) . This proves (REF ).
Finally, from Lemma REF  we have
\(\lim _{M \rightarrow \infty } \bar{\rho }_\mathbb {E}(S_M(F,G)) = 0\) 
and thus
\(\lim _{M \rightarrow \infty } \bar{\rho }_\mathbb {E}\left(\mathfrak {o}^{d+1} \bigcap \left(\bigcup \limits _{\mathfrak {p}\succ M} \widetilde{U}_\mathfrak {p}\right)\right) = 0.\) 
Thus, applying Theorem REF  and using (REF ), we get
\(\rho \left(\widetilde{\Gamma }^d\right) = 1-m(\lbrace \emptyset \rbrace )&= 1-\prod \limits _{\mathfrak {p}\in \mathcal {P}_K}(1-\nu _{\mathfrak {p}}(\widetilde{U}_\mathfrak {p})) \\&= 1-\prod \limits _{\mathfrak {p}\in \mathcal {P}_K}\left(1-p^{\deg (\mathfrak {p})}\cdot \dfrac{(p^{\deg (\mathfrak {p})}-1)^2}{p^{\deg (\mathfrak {p})(d+2)}}\right) \\&=1-\prod \limits _{\mathfrak {p}\in \mathcal {P}_K}\left(1- \dfrac{(N(\mathfrak {p})-1)^2}{N(\mathfrak {p})^{d+1}}\right).\) 
Now we turn to the case \(d=2\) . We use the same strategy as in [1]} and fix some positive integer \(M\)  and some integral basis \(\mathbb {E}\)  of \(\mathfrak {o}\) . Then we consider the system
\(V_\mathfrak {p} = {\left\lbrace \begin{array}{ll} \widetilde{U}_\mathfrak {p},& \mathfrak {p}\preceq M,\\ \emptyset ,& \text{else}, \end{array}\right.}\) 

and \(V_\infty = \emptyset \) . The system \((V_\eta )_{\eta \in N_K}\)  clearly satisfies the conditions of Theorem REF  and thus
\(1\ge \underline{\rho }_\mathbb {E}(\widetilde{\Gamma }^2)\ge \underline{\rho }_\mathbb {E}\left( \bigcup _{\eta \in N_K} V_\eta \cap \mathfrak {o}^{3} \right)&= 1 - \prod _{\mathfrak {p}\in \mathcal {P}_K, \ \mathfrak {p}\preceq M} \left( 1 - \frac{(N(\mathfrak {p})-1)^2}{N(\mathfrak {p})^3} \right) \nonumber \\&\ge 1 - \prod _{\mathfrak {p}\in \mathcal {P}_K, \ \mathfrak {p}\preceq M} \left( 1 - \frac{1}{2N(\mathfrak {p})} \right). \) 

By [2]} the series \(\sum _{\mathfrak {p}\in \mathcal {P}_K} N(\mathfrak {p})^{-1}\)  diverges to infinity. Hence, the product of (REF ) goes to zero for \(M \rightarrow \infty \) . Thus, we get
\(\rho (\widetilde{\Gamma }^2)=1 = 1- \prod _{\mathfrak {p}\in \mathcal {P}_K} \left(1- \frac{(N(\mathfrak {p})-1)^2}{N(\mathfrak {p})^{3}} \right).\) 
